TRANSFORMATIONS OF MATHEMATICAL · transformations of mathematical and stimulus functions chris...

TRANSFORMATIONS OF MATHEMATICALAND STIMULUS FUNCTIONS

CHRIS NINNESS

STEPHEN F. AUSTIN STATE UNIVERSITY

DERMOT BARNES-HOLMES

NATIONAL UNIVERSITY OF IRELAND, MAYNOOTH

AND

ROBIN RUMPH, GLEN MCCULLER, ANGELA M. FORD, ROBERT PAYNE,SHARON K. NINNESS, RONALD J. SMITH, TODD A. WARD, AND MARC P. ELLIOTT

STEPHEN F. AUSTIN STATE UNIVERSITY

Following a pretest, 8 participants who were unfamiliar with algebraic and trigonometricfunctions received a brief presentation on the rectangular coordinate system. Next, theyparticipated in a computer-interactive matching-to-sample procedure that trained formula-to-formula and formula-to-graph relations. Then, they were exposed to 40 novel formula-to-graphtests and 10 novel graph-to-formula tests. Seven of the 8 participants showed substantialimprovement in identifying formula-to-graph relations; however, in the test of novel graph-to-formula relations, participants tended to select equations in their factored form. Next, wemanipulated contextual cues in the form of rules regarding mathematical preferences. First, weinformed participants that standard forms of equations were preferred over factored forms. Ina subsequent test of 10 additional novel graph-to-formula relations, participants shifted theirselections to favor equations in their standard form. This preference reversed during 10 moretests when financial reward was made contingent on correct identification of formulas in factoredform. Formula preferences and transformation of novel mathematical and stimulus functions arediscussed.

DESCRIPTORS: value, preference, mutual entailment, combinatorial entailment, trans-formation of function

_______________________________________________________________________________

Many mathematical functions have graphsthat are transformations of a library of functionson Descarte’s rectangular coordinate system.Historically, transformation of graphs of func-tions has been a major component of manylevels of algebra as well as of more advancedcourses in mathematics. Showing students howa variable changes in defined stages has beena dynamic learning tool in training mathema-

tical relations. In other words, allowing studentsto ‘‘see’’ the transformations that occur when anequation is modified in its particular character-istics (Larson & Hostetler, 2001) helps thelearner to understand families of functions andtheir relations to each other. This can beparticularly beneficial in showing how transfor-mations are applied to mathematical functionson the coordinate axis, and how the equation ofa function can be systematically modified totransform the graph of any function (Sullivan,2002).

Nevertheless, instructional strategies aimed attraining transformation of mathematical func-tions have been circumvented in many highschool and college intermediate algebra classes(R. Huettenmueller, personal communication,

Portions of this paper were presented at the 31st annualconvention of the Association for Behavior Analysis,Chicago, May 2005.

Correspondence concerning this article should bedirected to Chris Ninness, School & Behavioral Psycho-logy Program, P.O. Box 13019, SFA Station, StephenF. Austin State University, Nacogdoches, Texas 75962(e-mail: [email protected]).

doi: 10.1901/jaba.2006.139-05

JOURNAL OF APPLIED BEHAVIOR ANALYSIS 2006, 39, 299–321 NUMBER 3 (FALL 2006)

299

June 25, 2005). This is somewhat surprising inthat, traditionally, many state assessment in-struments of algebra have included questionsregarding transformation of functions (e.g.,Texas Education Agency, 2002), and mostAlgebra I textbooks (e.g., Kennedy, McGowan,Schultz, Hollowell, & Jovell, 1998; Saxon,1997) and intermediate algebra textbooks(e.g., Larson & Hostetler, 2001) emphasizethe importance of exposure to these concepts.Moreover, this subject matter is often addressedin varying degrees of complexity within collegealgebra (e.g., Larson & Hostetler, 1997),trigonometry (e.g., Smith, 1998), precalculus(e.g., Sullivan, 2002), and calculus (e.g.,Finney, Weir, & Giordano, 2001) textbooks.Concurrently, math performance in the UnitedStates has been in a state of continuous lagbehind most of the countries in the industrial-ized world. Findings from the Programme forInternational Student Assessment (PISA, 2003)study indicate that 15-year-old students in theUnited States are performing at a disappointinglevel in fundamental math concepts whencontrasted with their counterparts in other partsof the industrialized world. For example, theUnited States ranked 24th of 29 countries withregard to mathematics literacy. Moreover, thePISA study indicated that 25% of Americanstudents performed below the lowest possiblelevel of competence in mathematics. This lackof fluency in the fundamentals of basicmathematical operations has migrated intopostsecondary education, where more thanone in three college students must enroll ina remedial math program prior to takingcollege-level courses (Steen, 2003).

From the perspective of behavior analysis, itseems likely that the omission of componentmath skills would contribute to a cumulativedysfluency in prerequisite and related problem-solving skills (see Binder, 1996, for a discussionof cumulative dysfluency). Consequently, ap-plied behavior-analytic methods for trainingtransformation of mathematical functions appear

to be needed, and indeed recent research hasbegun to address this issue (Ninness, Rumph,McCuller, Harrison, et al., 2005; Ninness,Rumph, McCuller, Vasquez, et al., 2005). Thiswork has drawn directly on some of theprocedures and concepts used in various typesof research on stimulus relations (e.g., Lane,Clow, Innis, & Critchfield, 1998; Leader &Barnes-Holmes, 2001; Lynch & Cuvo, 1995)and in particular on relational frame theory(RFT; Hayes, Barnes-Holmes, & Roche, 2001).One of the core postulates of RFT is that muchof human relational responding, includingmathematical reasoning, is established in theform of generalized relational operants throughappropriate histories of multiple-exemplar train-ing (e.g., Y. Barnes-Holmes, Barnes-Holmes,Smeets, Strand, & Friman, 2004). In fact, a widerange of new training protocols using stimulusrelations are beginning to appear in the appliedliterature. For example, Rehfeldt and Root(2005) confirmed that a history of relationalresponding generated requesting skills among 3adults with severe developmental disabilities. Inaddition, 1 of the participants demonstratedtacting and textual behavior. Rehfeldt and Rootspeculate that establishing derived relations hasthe potential to generate novel forms of request-ing and perhaps a wide range of complex verbalskills. According to RFT, there are many suchrelational operants (or relational frames), butthey all possess three defining behavioral prop-erties: mutual entailment, combinatorial entail-ment, and the transformation of stimulusfunctions. It is important that the latter shouldnot be confused with the mathematical variety,transformations of graphs of functions. Bothtypes of transformations are detailed below.

The concept of mutual entailment refers tothe derived relations that may obtain betweentwo stimuli or events. For example, if a givenstimulus is related to another such that StimulusA is the same as Stimulus B, then the derivedrelation—B is the same as A—is mutuallyentailed. The concept of combinatorial entail-

300 CHRIS NINNESS et al.

ment refers to derived relations among three ormore stimuli. For example, given that A is theopposite of B and B is the opposite of C, Csame as A and A same as C are defined ascombinatorially entailed relations. Amongmany other possibilities (e.g., greater than orless than), it might be that A is the same as Band B is the same as C, in which case A remainsthe same as C (and vice versa) as combinato-rially entailed. There are virtually an unlimitednumber of ways in which stimuli or eventsmight be mutually or combinatorially entailed,and many of these may result in comparisonsother than sameness. Nevertheless, in all cases,mutually and combinatorially entailed relationsconstitute a relational network, and when sucha network has particular stimulus functions, thefunctions of other events in that network maytransform or alter in accordance with thederived relations.

Transformation of Stimulus Functions

The concept of transformation of stimulusfunction refers to the changes that occur in thebehavioral functions of stimuli based on theirparticipation in a particular relational network.In the present study, transformation of stimulusfunction addresses how networks of newlylearned relations may become preferred overothers. As a practical matter, consider thefollowing brief example from Stewart, Barnes-Holmes, Hayes, and Lipkens (2001). Perhapsan English-speaking child has learned thata nickel has a particular type of financial valuein relation to a dime,

and furthermore has learned the arbitrary compar-ative relation of size, such that when offered a nickelor a dime she will avoid the physically larger nickelto pick the arbitrarily ‘‘larger’’ dime. If when visitingthe Netherlands the child is provided contextual cuesfor the derivation of a relation between a ‘‘stuiver’’and a ‘‘dubbeltje,’’ that is analogous to that betweena nickel and a dime (e.g., ‘‘stuiver is to dubbeltje asnickel is to dime’’) then she may now derive thata ‘‘stuiver’’ is half the value of a ‘‘dubbeltje.’’ (p. 77)

Conversely, if in the above example, the childwere to be informed (incorrectly) that the

relation between ‘‘stuiver’’ and a ‘‘dubbeltje’’was the ‘‘opposite’’ of that of similar coins in hernative land, she might well derive that a ‘‘stuiver’’is twice the value of a ‘‘dubbeltje.’’ Moreover,while still operating under the control of such aninaccurate rule, if given an opportunity to selectone of the two coins, this same individual is likelyto show a preference for the physically larger‘‘stuiver.’’ Here, the specific value function of thecoins transforms in accordance with their re-spective contextual cues (verbal descriptions oftheir relative value) and independent of anydirect training or reinforcement addressing themonetary systems involved.

In the same way that a particular discriminativestimulus may control a class of unique responsesin alternating contexts, a particular rule thatdescribes a context may alter a class of responses,but only in the context defined by the rule.Informing a child that when she is in a particularcountry, coins are the same as or the opposite ofsimilar coins in her native land may control herpreference for those coins, but only for theduration of her stay in that country. The rulesfunction as a form of instructional control, butonly conditionally. Although the above account isonly a theoretical example, it has practicalimplications for training several types of mathe-matical relations. The current study focused onthe transformation of preference functions forspecific relational or mathematical networks.

Transformation of Mathematical Functions

When graphs of functions change by theaddition (or subtraction) of terms withinformulas for those functions, they are said totransform. Collectively, these techniques areoften described as transformations of graphs offunctions (Larson & Hostetler, 2001). Critically,the concept of relating relations may beimportant in developing a behavior-analyticunderstanding of the transformation of math-ematical functions. When a student successfullylearns to relate a particular graph to a particularformula, for example, this involves more thansimply relating two discrete stimulus events.

TRANSFORMATIONS OF FUNCTIONS 301

Both the graph and the formula are composedof multiple stimuli that relate to each other inspecific ways, so relating graph to formula thusinvolves relating one set of relations to anotherset of relations.

For instance, many functions have graphsthat interrelate in their various types of verticaland horizontal shifts, reflections, compressions,and stretches. As one example of a mathematicaltransformation, the addition of a positiveconstant inside of the argument shifts thefunction to the left (negative direction), whereasthe subtraction of a constant inside of theargument produces a shift to the right (positivedirection). For many students, horizontaltransformations run contrary to expectation,and many students have difficulty learning toidentify graphed representations of such trans-formations when displayed individually or inmultiple combinations of horizontal and verti-cal shifts. Pilot research in our laboratorysuggests that horizontal shifts are especiallydifficult for many students when the argumententails a negative x coefficient and a negativeconstant (e.g., y ~

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

{x { 4p

). However, ourprevious investigations have demonstrated thattraining students to factor via matching-to-sample (MTS) procedures allowed them tolearn horizontal shifts more efficiently and toderive more complex transformations fora much wider range of formulas and graphedanalogues.

For example, Ninness, Rumph, McCuller,

Vasquez, et al. (2005) employed a series of

MTS procedures in training formula-to-graph

relations. Participants were tested on 36 novel

variations of the original equations, and most

demonstrated substantially improved perfor-

mance. Participant error patterns were identi-

fied with the help of an artificial neural network

system called the self-organizing map (Koho-

nen, 2001). Subsequently, revised software was

developed to remediate errors, and participants

showed improved performances in identifying

mathematical relations. In a follow-up study

(Ninness, Rumph, McCuller, Harrison, et al.,2005), participants took part in an MTSprocedure in which they received training onparticular standard formula-to-factored formulaand factored formula-to-graph relations as theseformulas pertain to reflections and vertical andhorizontal shifts. Participants demonstratedcombinatorial entailment by identifying stan-dard formula-to-graph relations and showedhigh levels of accuracy in identifying 40 noveltransformations of graphs of mathematicalfunctions. The fact that the participants werecapable of identifying novel transformationsindicated that they had not simply learned torelate formula and graphs as discrete stimulusevents. Instead, they had learned to relate therelations contained within the graphs andformulas to each other, and this relating ofrelations was then applied to the novel trans-formation problems. The present study soughtto extend this research.

In most mathematical textbooks, equationsof functions are illustrated in standard form,starting with the highest degree term andcontinuing with terms in descending order.(Depending on the function, some texts refer tothis as the general form of the equation.) Incontrast, factored equations are written as theproducts of lower degree expressions, and theyare most often used for explanatory purposes orto demonstrate the derivation of variousmathematical relations (Sullivan, 2002).

To delineate the present study from ourprevious work in this area, in the presentinvestigation we attempted to expand ouranalysis to include novel graph-to-formularelations as well as the transformation ofstimulus functions of formulas. Unlike mostlaboratory research in the area of derivedstimulus relations (e.g., Dougher & Markham,1994; O’Hora, Roche, Barnes-Holmes, &Smeets, 2002), the current investigation at-tempted to address contextual control bymanipulating verbal instructions that addressnovel mathematical relations. First, as in our


previous research (Ninness, Rumph, McCuller,Harrison, et al., 2005; Ninness, Rumph,McCuller, Vasquez, et al., 2005), participantswere given a brief lecture, followed by comput-er-interactive training on the relations betweenstandard formulas and more easily understoodfactored formulas as well as factored formulasand their graphical representations. MTS pro-cedures were used to assess mutual entailmentand combinatorial entailment, and participantswere probed over an array of novel and complexformula-to-graph relations. However, unlikeany of our previous studies in this area,participants also were probed on 10 novelgraph-to-formula relations, in which two of thesix comparison items were correct. One of thecorrect formulas was arranged in standard formand the other in factored form. Then, weattempted to transform participants’ preferencefor particular forms of correct answers byinforming them that mathematicians prefer toexpress equations in standard rather thanfactored form. Finally, we implemented a con-trol condition in which the experimentalcontingencies were altered by informing partic-ipants that they would receive increased finan-cial reward for selecting correct equationsrepresented in the factored form rather thanthe standard form.

METHOD

Participants and Setting

After obtaining informed consent and ad-ministering a pretest to determine individuals’familiarity with various mathematical functions,the experimenters dismissed anyone who dem-onstrated any prior knowledge of transforma-tion of mathematical functions. Correctlyanswering more than three of 15 pretest itemsprecluded an individual from participating inthe experiment.

Ten participants (9 women and 1 man),ranging in age from 19 to 35 years, began theexperiment, but 2 did not complete the entiretraining and testing sequence. All participants

were undergraduates or graduate students atStephen F. Austin State University. Participantswere recruited from various academic programsby way of prearranged agreements with profes-sors to offer this opportunity to their classes. Allparticipants received five test points on theirfinal examination for taking part in theexperiment. There were 70 tests of novelrelations in total. Each participant was paida maximum of $8.00 for the entire experiment,with 10 cents per correct response during theassessment of novel relations (i.e., the first 60items) and 20 cents per correct response duringthe last 10 trials. After completing the study, allparticipants were debriefed and compensatedaccording to the number of correct responsesthey provided in the assessment of novelformula-to-graph and graph-to-formula rela-tions. All sessions were conducted in rooms ofthe university that remained free of externaldiversions throughout the course of the exper-iment.

Apparatus and Software

Participants were seated individually at a tablein a small session room containing a HewlettPackard Pavilion ze5170 (Pentium 4 2-GHzprocessor with 512 MB RAM) laptop computerthat displayed both mathematical equations andgraphical representations of equations on a whitebackground. A Labtec AM 252 microphoneadjacent to each computer was conspicuouslyattached to a side port of the computer.Training and MTS procedures and the re-cording of responses were controlled by thecomputer program, which was written by thefirst author in Visual Basic 6 and C++. Thesoftware provided math instructional tutorials,displayed graphs, and recorded the accuracy ofresponses throughout all phases of the study.The experimental sessions were conducted onthe laptop with an attached infrared mouse.

Design and Procedure

After a brief pretraining presentation (Stage1), the participant was taken to the computer in


an individual treatment room for Stage 2, wherecomputer-interactive training was conducted.As in Ninness, Rumph, McCuller, Harrison, etal. (2005), participants were trained and testedon A-B and B-C relations and the assessment ofmutually entailed (B-A and C-B) and combi-natorially entailed (A-C and C-A) relations. Thecomputer program then assessed the participanton 40 novel relations between formulas and thegraphs of these functions.

At the conclusion of the assessment of novelformula-to-graph relations, participants’ prefer-ences for factored and standard forms of theequations were assessed. In this phase, weattempted to produce an experimental analogueof the transformation of stimulus functions, orchange in formula preference, by way ofproviding contextual cues in the form of verbalrules that described a mathematical preferencefor particular types of formulas over others. Inattempting to examine the transformation ofstimulus functions, rules were used to establishcontextual control over the discrimination ofcorrect formulas from the comparison array ofincorrect formulas. Participants were exposed tocomparison items with four incorrect formulasand two correct formulas. One of the correctformulas was represented in standard form andthe other in factored form. After participantsresponded to 10 graph-to-formula comparisons,they were given verbal rules indicating thatequations of graphs are preferred when they arerepresented in their standard form. Subsequentto responding to a set of 10 graph-to-formulacomparisons, we implemented a control condi-tion in which we told participants that theywould receive supplemental reinforcement(20 cents rather than 10 cents) for selectingthe correct comparison formula in factoredform rather than standard form.

Stage 1: Pretraining on basic mathematicalrelations. The experimenter provided a pretrain-ing lecture to participants individually usingPowerPointH illustrations of the rectangularcoordinate system. As illustrated in Figure 1,

the experimenter read rules regarding squareroot operations from the screen and answeredquestions, but only to the extent that thequestions pertained directly to sample-to-com-parison items (A-B or B-C). The relations of B-A, C-B, A-C, and C-A were not addressed.

Step 1: Provide A-B rules. In pretraining A-Brules, the experimenter explained that a negativecoefficient of x inside the argument of thestandard form of a formula often makesgraphing the function more complicated andthat the standard form of the formula can befactored to remove any negative signs thatpreceded the x variable. It was explained thatfactoring the standard formula makes it moreconducive to graphing techniques.

In this narrative account of A-B relations,standard formulas were samples, and factoredformulas served as comparisons. For instance,participants were shown a basic square rootfunction in its standard form and how toexpress it when a negative one coefficient isfactored out of the argument. In the pretrainingand computer training phases, we used thewords ‘‘negative sign’’ rather than ‘‘21 co-efficient’’ because most of our participants hadno familiarity with the latter term.

Step 2: Provide B-C rules. Because partici-pants were unfamiliar with the Cartesiancoordinate system, the experimenter noted thatthe horizontal number line is identified as the xaxis and that the vertical number line is calledthe y axis. Participants were told that varioustypes of formulas were called functions andcould be employed to produce graphs. UsingPowerPointH slides, the experimenter read rulesexplaining that a negative sign inside the radicalor the parentheses reflects the graph over in(about) the y axis and that a negative signoutside the radical or the parentheses reflects thegraph down in (about) the x axis. Likewise,a constant value added or subtracted outside theradical or the parentheses moves the graph up ordown the y axis in the same direction asindicated by the sign. The illustrations and rules


provided on the slide presentation were identi-cal to those of the computer-interactive MTSprocedures that followed this presentation.

Step 3: Provide A-B exemplar. Participantswere exposed to an instance of an A-B test screen.In this illustration, A was represented asa standard square root formula and B as one ofsix factored square root formulas, or compar-isons. The experimenter told participants thatthe same type of screen image would be usedduring the computer-interactive training session.Using a handheld laser pointer, the experimenteridentified the correct comparison item.

Step 4: Provide B-C exemplar. Participantswere exposed to a slide of a B-C test screen.

Here, B represented the factored formula, andC was one of six possible graphical representa-tions of B. Again, using the laser pointer, theexperimenter identified the correct comparisonitem displayed on the screen. (The A-B and B-CPowerPointH illustrations were not the same asthose used in Stage 2 for the actual computer-interactive training.)

As in Ninness, Rumph, McCuller, Harrison,et al. (2005), all participants were given thesame 15-min pretraining presentation regardingthe basics of the rectangular coordinate systemand the relation between the square rootformula and its graphical representation. Fol-lowing the presentations, participants were

Figure 1. Stage 1 included giving rules and rules with exemplars by way of a PowerPointH presentation. Stage 2involved computer-interactive conditional discrimination MTS training.


escorted to the session room, and the experi-menter demonstrated a point-and-click re-sponse on a sample screen and made sure thatthe participants could perform this type ofresponse independently. Participants were toldthat mathematical rules would be posted onvarious screens and were directed to read theserules into the microphone adjacent to thecomputer each time the program instructedthem to do so.

Stage 2: Training and testing of mathematicalrelations. Before beginning the experiment, allparticipants were advised that at various pointsduring the session, the computer would instructthem to stop responding. At that time, theywere to contact the experimenter, who waslocated in an adjacent office. Stage 2 involvedconditional discrimination MTS training; theprogram presented the participants with visualdisplays in the form of mathematical rules andemphasized that these rules should be statedaloud by reading them into a microphone.Thus, participants read and practiced thebehaviors specified by mathematical rules. Aftertraining and assessing standard formula-to-factored-formula (A-B) relations and factored-formula-to-graph (B-C) relations, participantswere tested for mutually entailed formula-to-formula (B-A) relations and graph-to-formula(C-B) relations. Then, the combinatoriallyentailed relations between the standard formof the formulas and their respective graphs (A-C), as well as the graphs of the functions andthe original standard formulas (C-A), weretested.

For samples and comparisons, formula sizewas set at 24 using Times New Roman font.When displayed as samples, graphs wereapproximately 2 in. square on the computerscreen; however, when displayed as comparisonsin groups of six, they were reduced toapproximately 1.5 in. square each. On anyoccasion during which a participant emitted anincorrect response, the program randomized allcomparison elements on all screens and reex-

posed the participant to the entire MTSprotocol, starting from the beginning.

Step 1: Train A-B relations, test A-B. As inNinness, Rumph, McCuller, Harrison, et al.(2005), computer-interactive training of A-Bincluded the following details. As the programdisplayed a mathematical rule on each trainingscreen, the computer instructed the participantto read this rule into the microphone. Forexample, the participant read a rule indicatingthat the standard form of the square rootformula can be factored to remove any negativesigns that precede the x variable. After recitingeach rule twice, the participant clicked ‘‘next’’to advance to the screen that assessed A-Bperformance (see Appendix A). Unlike theprevious training procedures employed inNinness, Rumph, McCuller, Harrison, et al.(2005), the computer program did not generateany audio output expressing the rules displayedon the screen, nor did we attempt to recordparticipants’ reading of the rules displayed onthe computer screen.

Step 2: Train B-C relations, test B-C. Thesame MTS procedure was used to train and testB-C relations but focused on rules pertaining tofactored equations and their graphs. Rulesregarding reflections and vertical and horizontalshifts associated with various functions on thecoordinate system were displayed and then readaloud by participants. For example, the screendisplayed solid blue lines to represent the basicsquare root functions and dashed red lines toillustrate transformations of the square rootfunctions that occurred when the formulaschanged. These rules described mathematicalrelations such as, ‘‘Negative sign inside theradical is reflected over in the y axis. A positiveconstant inside the radical or the parenthesesmoves the function in the opposite directionalong the x axis.’’ (The program used the words‘‘over in’’ rather than the conventional ‘‘about’’based on pilot testing feedback that indicatedthis phrasing was easier for many participants tounderstand. See Appendix B for an illustration.)


After these rules were displayed twice, theparticipant clicked ‘‘next’’ to advance to thescreen that assessed B-C performance. Althoughthe assessment screens varied the specific valuesof the constants within the formulas, the correctanswers were always obtainable by performingin accordance with the previously displayedrules.

Step 3: Test B-A, test C-B, test A-C, and testC-A. This step assessed the mutually entailedB-A and C-B relations and the combinatoriallyentailed relations between the graphs of thefunctions and the original standard equations(A-C). Also, the combinatorial relations betweenthe standard forms of the formulas and theirgraphs (C-A) were tested (see Appendix C).

All mathematical rules, formulas, and graph-ical representations for horizontal and verticalshifts as well as for reflections in the x axis and yaxis were trained according to this protocol. Intotal, the program trained and tested twoversions of the square root function and twoversions of the common logarithmic function.

Figure 2 illustrates the trained and assessedrelations addressing the log function A2, B2,and C2. Following the trained (A-B and B-C)relations, the program tested the mutuallyentailed (B-A and C-B) relations and combina-torially entailed (A-C and C-A) relations.

Correction procedures and mastery criteria.Mastery of the basic mathematical relationsrequired the participant to complete one error-less sequence of all four classes of relatedmathematical functions. This included sixMTS tests (A-B, B-C, B-A, C-B, A-C, andC-A) within each class of functions. Therefore,mastery required 24 consecutive correct identi-fications of the formula-to-graph and graph-to-formula relations. Participants who made MTSerrors during the assessment of relations werereturned to the beginning of the program; theprogram randomized all comparison elementsbefore reexposing participants to the MTSprotocol. If a participant required more thanfive reexposures to the protocol, the program

ended, and the participant was compensated,debriefed, and excused from the study.

Assessment of novel formula-to-graph relations.After demonstrating mutual entailment andcombinatorial entailment on both versions ofthe square root functions and both versions ofthe common logarithmic functions, participantswere tested on novel relations between formulasand their graphs—specifically, 40 complexvariations of the trained mathematical formu-la-to-graph relations. For each test item, theparticipant attempted to match a new formulawith a graph from a selection of six graphs notpreviously used during any training andassessment screens. (Within any MTS test, thecomparison graphs were all the same type ofmathematical function; for instance, a sineformula was only compared against variationsof sine graphs.) No accuracy feedback orreexposure to training was given during assess-ment of novel formula-to-graph relations. Thistask required participants to identify graphsrelating to formulas with new combinations ofpositive and negative constants inside andoutside the arguments of 40 new mathematicalfunctions. These novel formula-to-graph assess-ments included multiple combinations ofreflections and vertical and horizontal shiftsfor complex logarithmic, square root, exponen-

Figure 2. A square root function where A2 is in

standard form, B2 is in factored form, and C2 is thegraphical representation of this function.


tial, square, cubic, tangent, and sine functions.Performing these MTS tasks did not require theparticipant to become familiar with the dynam-ics of each type of mathematical function.However, successful performance did requirethe participant to identify new and complexcombinations of reflections and shifts as theyoccurred among a wide array of diversifiedfunctions. In effect, this task involved relatingspecific relations expressed within the formulato specific relations illustrated within thegraphs.

Figure 3 illustrates one of the 40 novelformula-to-graph tests. The solid line on eachof the six comparison graphs represents thebasic function prior to transformation. Thedashed line shows the graph produced by thenovel formula. When given the formula fora cube function with a negative constant 4inside the argument (shifting right) and a neg-ative 4 outside the argument (shifting down),participants who answered correctly chose C asthe mathematical transformation of this func-tion.

Figure 3. One of the 40 tests of novel formula-to-graph relations. The solid lines represent the basic cube function (y5 x3), and the dashed lines indicate the possible transformation when the formula becomes more complex. A participant

who identified a novel variation of the formula that included a negative constant 4 within the argument and a negativeconstant 4 following the argument would select C as the correct comparison item.


Assessment of novel graph-to-formula relations.After providing 40 formula-to-graph compar-isons in the format illustrated in Figure 3, weassessed participants’ preferences for factoredand standard forms of the equations. Here, theprogram provided 10 trials of novel graph-to-formula comparisons. On each screen, a graph-ical representation of a function was positionedat the top and an array of six equations wasdisplayed below. In each array, two of thecomparisons were correct: one in factored formand the other in standard form. Figure 4 shows1 of the 10 screen displays employed during thisphase, where F represents the standard form andD represents the factored form of the argumentof this function.

Identifying standard arguments. After theparticipant responded to the 10 graph-to-formula comparisons, the computer interruptedthe program with the following message:

‘‘Please stop and contact the experimenter.OK?’’ After being summoned, the experimentergave the participant a brief statement regardingthe types of equations preferred by mathema-ticians and others working in related academicareas. While pointing to a graph and itsequation within a precalculus textbook (Sulli-van, 2002), the experimenter stated,

Generally, standard formulas of graphs are consid-ered to be more elegant and sophisticated since theycontain fewer terms and operations. In the next partof the experiment, there could be more than oneform of the correct answer on some tests. Mathe-maticians try to identify equations in their standardform if they are available.

The participants then clicked ‘‘OK’’ on theinput box of the computer screen and pro-ceeded to respond to another series of 10 novelgraph-to-formula comparisons in which two ofthe comparison items were correct. Again, oneof the correct answers was in standard form andthe other was in factored form.

Identifying factored arguments. After theparticipant responded to the set of 10 graph-to-formula comparisons, the program againinterrupted the testing procedure by displayingthe message, ‘‘Please stop and contact theexperimenter.’’ After being summoned, theexperimenter gave the participant the followingrevised statement regarding the experimentalcontingencies: ‘‘This time you can double yourearnings. We will give you 20 cents each timeyou identify the correct formula in its factoredform.’’ This phase entailed 10 more tests ofgraph-to-formula relations with two mathemat-ically correct equations within each set of sixcomparisons. Again, one of the two correctequations in the array of comparisons was thestandard form and the other was the factoredform. At the conclusion of the 10th test, theprogram ended, and the participant was de-briefed and reimbursed according to thenumber of comparison items correctly identi-fied during the assessment of all novel relations.Figure 5 illustrates the first 10 formulas used inassessing novel graph-to-formula relations, dis-

Figure 4. One of the 30 tests of novel graph-to-formula relations. The solid line represents the basic

exponential function (y 5 x3), and the dashed lineindicates the transformation when the formula becomesmore complex. A participant who identified that this

function included a negative coefficient of x, a positiveconstant 6 within the argument, and a positive constant 4following the argument would select F as the correctcomparison item in its standard form or D as the correct

comparison item in its factored form.


played in standard and factored forms. Theremaining 20 formulas were variations of thesame types of formulas listed in this figure.

RESULTS

One of the participants made a series oferrors during the MTS training procedure. Shewas reexposed to training five times, after whichthe computer automatically exited her from theprogram. She was debriefed, compensated, andexcused from the study. A 2nd participant wasinterrupted during her first exposure to thetraining sequence and was unable to return dueto a scheduling conflict. The remaining 8participants completed the MTS training pro-cedure in fewer than five exposures. Table 1shows the number of required exposures foreach participant to complete the training of allA-B and B-C relations.

Figure 6 shows the 40 probe formulasemployed in the test of novel formula-to-graphrelations. The subscripts of y identify thesequence in which these formulas were pre-sented by the computer program. For example,y20 5 log(x + 4) was the 20th item given in theassessment of novel formula-to-graph relations.All but one of these test items are identical tothose used in the Ninness, Rumph, McCuller,Harrison, et al. (2005) study.

The top of Figure 7 shows the pretest errorpatterns of our eight participants on 15formula-to-graph test items. Although the first15 items in the pretest were reemployed, all thecomparison items were randomized during the

Figure 5. The first 10 formulas as displayed instandard (left) and factored forms (right). Two additionalsimilar sets of 10 formulas in standard and factored forms

were used during the experiment.

Table 1

Number of Training Exposures Required to Attain Mastery

Participant A1-B1 B1-C1 A2-B2 B2-C2 A3-B3 B3-C3 A4-B4 B4-C4 Total

1 2 2 2 2 1 1 1 1 122 3 3 3 3 3 3 3 1 223 3 3 3 3 3 1 1 1 184 2 2 2 2 2 2 1 1 145 1 1 1 1 1 1 1 1 86 3 3 2 2 2 2 2 2 187 2 2 2 2 2 1 1 1 138 3 3 3 3 2 2 2 2 20

Figure 6. The 40 probe formulas employed in the test

of novel formula-to-graph relations. The subscripts of yidentify the sequence in which these formulas werepresented by the computer program.


computer-interactive test of novel relations thatfollowed MTS training. Items 16 through 40were composed of 25 novel formulas. Partici-pants were not exposed to any of these formulas(or their graphs) during any part of theirtraining or pretraining. Pretest results appearto confirm that the participants were unfamiliarwith the types of formula-to-graph relationsused in this experiment.

Error Pattern 1 includes 3 participants whodemonstrated above 92.5% accuracy during theassessment of novel stimulus relations. All 3failed to respond correctly to the 11th test item(y11 in Figure 6), which included a leading 21coefficient before the argument of a squarefunction. These participants shared no othercommon errors in the testing sequence.

Error Pattern 2 shows 2 participants whocorrectly responded to the novel mathematicalrelations with an accuracy level at or above82.5%. They shared errors on Novel Test Items

6, 14, 15, 32, and 40. Item 6 requiredidentification of a graph when its formulaincluded a negative square root with a negativeargument. Item 14 involved a novel exponentialequation containing the number 4 in the baseand exponent. Item 15 was very similar butinvolved an exponential equation containingthe number 4 in the base and a constant 4subtracted outside the argument. Item 32involved a square root function with a negativeargument and a constant 4 added outside theargument. Item 40 addressed a novel cubefunction in standard form. This functionincluded a leading negative coefficient of xand a negative constant 4 inside the argument.This item was also missed by 1 otherparticipant, who was not classified in this errorpattern.

Error Pattern 3 shows 2 participants whomade several errors consistent with those seen inError Pattern 2. These participants also missed

Figure 7. The top block shows the correct and incorrect responses on the pretest. Problem numbers are listed along

the x axis for each of the 8 participants. Accurate responses contain the digit 0; errors are shaded blocks containing 1. Thefour blocks beneath show the same participants’ error patterns following training and classification by error patterns.


Item 16, which was arranged in a similarexponential format. In addition, both partici-pants failed to correctly identify the novelformula-to-graph relation in Items 3, 16, 17,30, and 34; however, there do not appear to beany unique or distinguishing characteristicsregarding these test items. Participants in thiserror pattern correctly identified novel mathe-matical relations with an accuracy level at orabove 75%. Surprisingly, all of the participantsin Error Patterns 1, 2, and 3 correctly identifiedItem 29, which included a double negative signpreceding the argument of a square function.

Error Pattern 4 includes only 1 participant,who showed more difficulty responding to thenovel formula-to-graph relations than the otherparticipants. This participant made errorsconsistent with those of other participants andseveral other errors that defied classificationwith other participants. Nevertheless, she didcorrectly identify 62.5% of the novel formula-to-graph relations. Table 1 shows that sherequired more training exposures than any ofthe other participants to obtain mastery. Otherthan this participant, there appears to be verylittle correspondence between the number ofexposures required to reach the mastery criteriaand the number of errors that occurred duringthe assessment of novel relations.

Thirty additional test screens (41 to 70)assessed transformation of novel graph-to-formula relations. These screens contained twocorrect formulas in the array of six comparisons.For each array of comparisons, one correctformula was presented in standard form and theother in factored form. Thus, for any given test,the probability of being correct was approxi-mately 33%; we did not attempt to categorizethese outcomes by analyzing error patterns. Thefirst 10 graph-to-formula tests were givendirectly after the tests of novel formula-to-graph relations. Participants were not informedthat we had included two forms of correctformulas on each test screen (as shown inFigure 4). Figure 8 shows that 7 of the 8

participants tended to select formulas in theirfactored form during the first 10 novel graph-to-formula relations. Participant 6 correctlyresponded to all 10 tests, selecting five formulasin factored form and five in standard form.Only Participant 2 demonstrated more thanone error during these tests of novel graph-to-formula relations.

After responding to the 10th test screen inthis series, the computer displayed a messageindicating that the participant was to stop andcontact the experimenter, who provided con-textual cues in the form of rules. Here,participants were informed that mathematicianshad a distinct preference for formulas expressedin standard form; subsequently, all 8 partici-pants demonstrated a conspicuous shift towardselection of formulas in standard form. How-ever, 5 of the participants still selected two orfewer formulas in their factored form. OnlyParticipant 2 made more than one error duringthese tests.

This trend shifted when the experimenterindicated that reinforcement would be doubledfor identifying formulas in their factored form.During the final 10 tests, only 3 participantsmade errors and 5 of the participants selectedfactored formulas exclusively. In general, for-mula selection was consistent with the contex-tual cues provided by the experimenter. Partic-ipants averaged 53 min (range, 43 to 67 min)to complete training, MTS testing, testing ofnovel formula-to-graph relations, and testing ofnovel graph-to-formula relations.

DISCUSSION

The current results replicate and extend thoseof our earlier investigations of the relating ofmathematical formulas to mathematical equa-tions (Ninness, Rumph, McCuller, Harrison, etal., 2005; Ninness, Rumph, McCuller, Vas-quez, et al., 2005). After completion of training,participants were exposed to 40 novel formula-to-graph tests and 30 novel graph-to-formulatests. Of the 8 participants who completed the


Figure 8. Participants tended to select formulas in their factored form during the first 10 novel graph-to-formularelations. All 8 participants tended to shift their selection preferences in favor of the standard forms of the formulas

following the manipulation of rules suggesting that mathematicians favored formulas expressed in their standard form.This trend shifted when the experiment indicated that reinforcement would be doubled for identifying formulas in theirfactored form.


MTS training sequence, 7 demonstrated sub-stantially improved performance in identifyingnovel formula-to-graph relations. Then, partic-ipants were exposed to a series of 10 novelgraph-to-formula tests. In each trial, six com-parison formulas included one correct versionof the equation in factored form and one correctversion in standard form. Our data suggest thatparticipants tended to select equations in theirfactored form. This outcome is not especiallysurprising in that the factored forms of theseequations require fewer operations to derive thetransformation of graphs of functions.

In the next stage, we manipulated two differentcontextual functions. First, we gave the partici-pants a brief lecture regarding the mathematicalimportance of being able to recognize equationsin their standard form and subsequently assessedtheir performance with 10 more tests of novelgraph-to-formula relations. Here, participantsdemonstrated a shift in preference and tendedto select comparison equations in their standardform. Then, we provided contradictory instruc-tions for the final 10 test items, with increasedfinancial reward contingent on identifying for-mulas in their factored form. Specifically,participants received 20 cents per correct prob-lem for identifying the factored form of themathematical relations (10 cents for the standardform). Participants exhibited a systematic prefer-ence for factored forms of the comparisonequations. Results confirm that participantschanged their formula preferences in accordancewith the contextual functions that were in effect.

Shifting Formula Preferences

Given that graphed and factored formulaswere paired during B-C training, selectinga standard formula in the presence of a graphentailed more complex combinatorial relationsthan selecting a factored formula. Therefore,participants’ initial preference for factoredforms of the formulas is not surprising.However, attention should be given to thenotable shift in preferences in favor of the moreevasive standard form of the formulas when

given specific rules stating that standardformulas of graphs are considered to be moreelegant and sophisticated. Likewise, the fact thatparticipants once again shifted their preferenceswhen given rules informing them that theycould double their earnings by simply identify-ing the correct formula in its factored formmerits consideration.

Just as preference for large versus small coinsmay change in accordance with the currencyrules established across different countries, so,too, might a student’s preference for particulartypes of mathematical formulas alter in accor-dance with the rules established across experi-mental or real academic settings. Indeed, thereare many instructors of advanced mathematicsclasses who would disapprove of expressingformulas in factored form (despite the fact thatthe factored forms might be more explanatory).

This begs the question as to whether verbalrules are sufficient to constitute a form ofcontextual control. It has been argued that oncea repertoire of relational framing has beenestablished, relational frames and networks canthen function as contextual cues for relationalframing itself (e.g., D. Barnes-Holmes et al.,2001). The finding that a transformation offunction occurs under the control of a rule (i.e.,relational network) is consistent with thisargument. These data, particularly the shifts inpreference from factored to standard and backto factored equations, demonstrate a transfor-mation of stimulus functions in the context ofa mathematical relating relations task. Stewart,Barnes-Holmes, Roche, and Smeets (2002)reported a broadly similar effect, but in thecontext of a basic experimental procedure thatdid not articulate directly with an educational,or indeed any, applied behavior-analytic con-cerns. The extent to which our participantscame under the immediate and flexible controlof contextual cues in the form of rules regardingthe value of standard formulas [e.g., y 5

2ln(2x + 6) + 8] or factored formulas [e.g., y5 2ln(2(x 2 6)) + 8] demonstrates how easily


such rules may alter newly established complexrepertoires. Whether or not participants selectedthe factored or standard versions of the correctanswers depended not only on the newlyderived repertoires for graph-to-formula rela-tions but also on verbal rules describing thechanging value of particular types of formulasacross experimental conditions.

The transformation of stimulus functions hasbeen demonstrated many times in the labora-tory environment (e.g., Stewart et al., 2002);however, in the laboratory, contextual control istypically established using novel arbitrary stim-uli (but see Kohlenberg, Hayes, & Hayes, 1991,who employed gender-specific names as con-textual cues). In the current study, we usednatural language as a means for establishingcontextual control. The finding that a trans-formation of function occurs under the controlof a rule (i.e., relational network) is consistentwith our view that once a repertoire of relationalframing has been established, relational framesand networks can then function as contextualcues. Insofar as this view is correct, there may beconsiderable benefit in conducting furtherstudies in which natural language and RFTtraining procedures are intermingled. Doing sowill allow researchers to determine if, and towhat extent, the RFT approach to complexhuman behavior is valuable in producing greaterprediction and influence in applied or naturalsettings.

If our participants had learned simply toidentify four three-member classes composed offormulas and graphs in the same manneremployed in laboratory arrangements usingcompletely arbitrary shapes or nonsense sylla-bles, then we might expect less robust perfor-mance levels in tests of novel mathematicalrelations. However, the formula-to-graph andgraph-to-formula relations identified by partic-ipants in this study were not directly compara-ble and allowed participants to identify net-works of relations extending from the exemplarsprovided during training. Our view, therefore,

is that participants were not relating samplesand comparisons as unitary stimuli but wererelating relations contained in the formulas tothe relations found in the graphs. Consequent-ly, it would be overly simplistic to argue that thecurrent data could be described simply in termsof four three-member equivalence classes (seeHayes & Barnes, 1997, for a related discussion).

Multiple Exemplars in Mathematics

Multiple-exemplar training and derivedtransformation of functions have produceda series of impressive results with childrenacross a wide range of skills (e.g., Y. Barnes-Holmes, Barnes-Holmes, Roche, & Smeets,2001), and perhaps it is not surprising that suchmethods produced similar effects in trainingmathematical relations. Nevertheless, multiple-exemplar training procedures that employa wider range of factoring techniques inmathematics require more extensive investiga-tion. For example, as noted in Ninness,Rumph, McCuller, Harrison, et al. (2005),a quadratic function may be expressed as y 5 x2

2 8x ; however, in this form, the equation doesnot immediately lend itself to graphing.Factoring (completing the square) allows oneto obtain y 5 (x 2 4)2 2 16. This version ofthe formula increases the likelihood thata participant who has learned to identifyformula-to-formula and formula-to-graph rela-tions, by way of the above experimentalexercises, may also identify y 5 (x 2 4)2 216as a graph that shifts horizontally to the right by4 units and vertically 16 units.

As alluded to earlier, the concept of relationalframes indicates that derived relations thatcould address a virtually infinite number ofdiversified relations within a network ofmutually entailed and combinatorially entailedrelations such as same as, greater than, less than,opposite of, inverse of, or reciprocal of. In ourlaboratory, we have recently explored recipro-cals functions in which, given that A is thereciprocal of B and B is the reciprocal of C, Csame as A and A same as C are derived as


combinatorially entailed. For example, a func-tion such as y 5 sin(x) has a reciprocal ofcosecant function, y 5 csc(x), and the graphs ofthese functions generate lines going in differentdirections along the coordinate axis. Taking thereciprocal of the cosecant function, we obtain y5 1/csc(x). Here, the reciprocal of a reciprocalis combinatorially entailed such that y 5 sin(x)is the same as y 5 1/csc(x), and the graphs ofthese functions appear precisely the same as theyform a Pythagorean identity (see Figure 9 for anillustration of the equations and their respectivegraphical functions). Note that only the formulasin Figure 9 actually serve as samples andcomparison items within our training protocols;however, the graphs below each formula help toillustrate the topographic changes produced bythe respective formulas within the relationalnetwork. We are developing several programs onour experimental Web site to train these types ofreciprocal functions (see www.sfasu.edu/hs/

TransSinCosCscSec06.html as one illustrationof this protocol in progress). Indeed, preliminaryfindings using similar types of RFT protocols aremost promising (Ninness et al., 2006). More-over, multiple-exemplar training in our labora-tory has shown that students who had difficultylearning transformations of graphs of functionswere able to acquire such relations followingrelatively brief MTS training.

As contextual cues in the form of newmathematical rules are incorporated into a re-lational network, most students only requireexposure to a minimal number of exemplarsconsistent with precise rules to derive andrespond appropriately to many more detailedand complex mathematical relations. We antic-ipate that these procedures may contribute tothe development of more sophisticated trainingplatforms that permit users to derive complexmathematical relations in less time and withgreater accuracy. As another variation on this

Figure 9. The function y 5 sin(x) has a reciprocal of cosecant function y 5 csc(x). Taking the reciprocal of thecosecant function, y 5 1/csc(x) is obtained, and in this example, the reciprocal of a reciprocal is combinatorially entailed

and y 5 sin(x) is the same as y 5 1/csc(x).


theme, Binder (1996) demonstrated that ap-plying frequency-building methods to compo-nents in an analogue of stimulus equivalenceexperiments produced improved acquisitionrates during MTS procedures. Similar outcomeshave been demonstrated repeatedly in precisionteaching studies (e.g., Van Houten, 1980).Frequency-building procedures could be in-corporated into our software as we developmore advanced versions of our training proto-cols.

Caveats

As in our earlier research in this area, thisstudy did not attempt to isolate the influence ofthe pretraining lecture from that of thecomputer-interactive MTS training procedures.Given that our participants were relatively naiveregarding graphing techniques as they appliedto the coordinate system, some introductorydiscussion was necessary for them to interactwith the training and testing system. With theresults from this study and those of our previousinvestigations, it is becoming increasinglyobvious that particular entry-level skills arerequired to initiate training on such platforms.Moreover, we are not suggesting that theseprocedures constitute a stand-alone instruction-al system for training all of the complexity ofmathematical functions. It does suggest, how-ever, that having learned to match factored andstandard formulas and having learned toidentify the graphs of these functions witha limited number of basic exemplars, partici-pants derived a much wider array of morecomplex and diversified formula-to-graph aswell as graph-to-formula relations.

Much of the current pedagogical emphasis inU.S. high school and college mathematicsinstruction is directed at facilitating students’discovery of preexisting mathematical concepts(Biggs & Moore, 1993). From kindergartenmath (e.g., Macmillan, 1990) through calculus(e.g., Dubinsky & Schwingendorf, 1991),learners are encouraged to uncover commonal-ities and differences in their cognitive schema as

they encounter all manner of mathematicalchallenges. From this perspective, studentsconstruct their own knowledge of mathematicalrelations and discover principles for themselves(Slavin, 2006). Notwithstanding the pedagogi-cal emphasis in math education, the currentlack of fluency in fundamental math skills hasnow saturated substantial proportions of thepostsecondary population, with more than onein three college students required to enroll inremedial math before being permitted takecollege-level courses (Steen, 2003).

One might fairly question whether a tech-nology based on derived relational respondingcan play a role in addressing such pervasivedeficits. Clearly, we have not yet demonstratedin any of our research to date that a history ofreinforced relational responding is superior tothe more traditional forms of math instruction,nor have we yet published evidence thata history of multiple-exemplar training is moreeffective or efficient than training all possiblemathematical relations directly. Nevertheless,many behavior analysts are focusing on thedevelopment of specific instructional strategiesthat employ multiple-exemplar training as itapplies to the critical features among relationsin mathematics. Used in conjunction witha well-designed mathematics curriculum, webelieve that an MTS approach to mathematicalrelations might be adapted to traditionalmodalities of instruction. Lynch and Cuvo(1995) as well as Leader and Barnes-Holmes(2001) have shown similar types of outcomeswith different populations and different typesof exemplars. With a current emphasis onbasic trigonometric functions, we are attempt-ing to develop and deploy a Web-based betaversion of multiple-exemplar training at www.sfasu.edu/hs/NinnessTrigGraph06.html. (Notethat although these links may be useful forpreliminary or complementary instruction re-garding amplitude and frequency of odd andeven trigonometric functions, a complete MTSprotocol is required to train these functions


most effectively.) Pilot research in our labora-tory that addresses several trigonometric rela-tions pertaining to amplitude and frequency (aswell as the reflections and shifts described inthis study) suggests that multiple-exemplartraining procedures can be very effective inrapidly generating a wide range of complexmathematical relations among individualswithout any history in this area. Preliminaryoutcomes suggest that our MTS protocolsproduce substantially higher acquisition ratesof formula-to-graph and graph-to-formulatrigonometric relations than the process ofdirectly training all possible combinations ofthe various math functions under consideration(e.g., Ninness et al., 2006). By expanding ourcurrent architecture to include more basicalgebra, trigonometry, and precalculus rela-tions, we hope to deploy a series of onlineprotocols directed at facilitating the remedia-tion of students who suffer the insidious effectsof cumulative dysfluencies (Binder, 1996). Betaversions of these applications are freely avail-able to interested users; however, a completeWeb-based platform allowing us to rigorouslycompare multiple-exemplar training to manyof the more traditional math education systemsis still under development.

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Received September 26, 2005Final acceptance February 22, 2006Action Editor, Ruth Anne Rehfeldt

APPENDIX A

Moving from the standard form of theformula to the most conducive form of theformula, the conducive form always factorsout any negative signs that may precede x.

APPENDIX B

Negative sign inside the radical is reflectedover in the y-axis

A positive constant inside the radical orthe parentheses moves the function in theopposite direction along the x-axis.

Moving from the standard form of the formula to the most conducive form of the formula,the conducive form always factors out any negative signs that may precede x.


APPENDIX B

Negative sign inside the radical is reflected over in the y-axisA positive constant inside the radical or the parentheses moves the function in the opposite

direction along the x-axis.


APPENDIX C


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